How to Join Nightingale Skyrim?

You are correct in that you will need to join the Thieves Guild. After running through the Guild's main questline, you will reach the quest Trinity Restored, in which your character is invited to join the Nightingales. The armour will be provided free as part of the initiation ceremony.

How do you become a nightingale agent in Skyrim?

In order to become a Nightingale, you must finish the Thieves' Guild quest line. Bear in mind that you will lose the Skeleton key when you do.

Can you get Nightingale armor without joining the Thieves Guild?

You'll also have to do most of the Thieves Guild questline anyway so unless you have an RPing issue with becoming a Nightingale you may as well become a Nightingale and finish out the questline to get the rest of the benefits.